What are the best hyper-parameters to tune in reinforcement learning?
Personally, I would choose the following two as the most important:

epsilon: When using an epsilon-greedy policy, epsilon determines how often the agent should explore and how often it should exploit. Balancing exploration and exploitation is crucial for the success of the learning agent. Too little exploration might not teach anything to the agent and too much exploration might just waste your time.
learning rate: The learning rate determines how fast do you learn from new states of experience. A learning rate that is too high might not be good in cases when the environment has many states with high probabilities of negative rewards, i.e. many penalizations. This might make your agent move back and forth in the same place in order to avoid getting penalized. Also, a learning rate that is too low might make your agent learn very slowly and depending on your epsilon, the agent might enter a phase of exploitation with very little knowledge of an optimal policy.